C-k-Reconstruction of surfaces from partial data

In this paper, we study the problem of constructing a smooth approximant of a surface defined by the equation z = f(x(1), x(2)), the data being a fini te set of patches on this surface. This problem occurs, for example, after geophysical processing such as migration of time-maps or depth-maps. The us ual algorithms to solve this problem are picking points on the patches to g et Lagrange's data or trying to get local junctions on patches. But the fir st method does not use the continuous aspect of the data and the second one does not perform well to get a global regular approximant (C-1 or more). A s an approximant of f, a discrete smoothing spline belonging to a suitable piecewise polynomial space is proposed. The originality of the method consi sts in the fidelity criterion used to fit the data, which takes into accoun t their particular aspect (surface's patches): the idea is to define a func tion that minimizes the volume located between the data patches and the fun ction and which is globally C-k. We first demonstrate the new method on a t heoretical aspect and numerical results on real data are given.